FREE GRADIENT DISCONTINUITY AND IMAGE SEGMENTATIONansobol/otarie/slides/tomarelli.pdf · This talk deals with free discontinuity problems related to contour enhancement in image segmentation,

Blake & Zisserman functionalEuler equations

FREE GRADIENT DISCONTINUITYAND IMAGE SEGMENTATION

Franco Tomarelli

Politecnico di MilanoDipartimento di Matematica “Francesco Brioschi”

[email protected]

http://cvgmt.sns.it/papers

Optimization and stochastic methodsfor spatially distributed information

Sankt Petersburg, May 13th 2010

Franco Tomarelli


joint research withMichele carriero & Antonio Leaci ( Università del Salento, Italy )

Franco Tomarelli


Franco Tomarelli


Abstract:This talk deals with free discontinuity problems related to contourenhancement in image segmentation, focussing on

the mathematical analysis of Blake & Zisserman functional,

precisely:

1 existence of strong solutionunder Dirichlet boundary condition is shown,

2 several extremal conditions on optimal segmentation are stated,3 well-posedness of the problem is discussed,4 non trivial local minimizers are analyzed.

The segmentation we look for providesa cartoon of the given image satisfying some requirements:the decomposition of the image is performed by choosing a pattern oflines of steepest discontinuity for light intensity,and this pattern will be called segmentation of the image.

Franco Tomarelli


Eyjafjallajökull

Franco Tomarelli


Franco Tomarelli


Franco Tomarelli


Franco Tomarelli


rotoscope

Franco Tomarelli


A classic variational model for image segmentation has beenproposed by Mumford & Shah , who introduced the functional∫

Ω\K

(|Du(x)|2 + |u(x)− g(x)|2

)dx + γ Hn−1(K ∩ Ω) (1)

where

Ω ⊂ Rn (n ≥ 1) is an open set,

K ⊂ Rn is a closed set,

u is a scalar function,

Du denotes the distributional gradient of u,

g ∈ L2(Ω) is the datum (grey intensity levels of the given image),

γ > 0 is a parameter related to the selected contrast threshold,

Hn−1 denotes n − 1 dimensional Hausdorff measure.

According to this modelthe segmentation of the given image is achieved byminimizing (1) among admissible pairs ( K , u ),

say closed K ⊂ Rn and u ∈ C1(Ω \ K ).Franco Tomarelli


This model led in a natural way to the study ofa new type of functional in Calculus of Variations:

free discontinuity problem.

Existence of minimizers of (1) was proven by

De Giorgi, Carriero & Leaci (1989)

in the framework of bounded variation functions without Cantor part(space SBV ) introduced in

De Giorgi & Ambrosio.

Further regularity properties of optimal segmentationin Mumford & Shah model were shown by

[Dal Maso, Morel & Solimini, (1992), n = 2, ]

[Ambrosio, Fusco & Pallara (2000)],

[Lops, Maddalena, Solimini, (2001), n = 2, ],

[Bonnet & David (2003), n = 2 ].

Franco Tomarelli


stair-casing effect

Franco Tomarelli


Franco Tomarelli


Franco Tomarelli


To overcome the problems and aiming to better description ofstereoscopic images they proposed a different functional includingsecond derivatives.

Blake & Zisserman variational principle faces segmentation as aminimum problem:

input is given by intensity levels of a monochromatic image,

output is given by

meaningful boundaries whose length is penalized(correspond to discontinuity set of the given intensity and of itsfirst derivatives)

a piece-wise smooth intensity function(smoothed on each region in which the domain is splitted bysuch boundaries).

Franco Tomarelli


another problem with free discontinuity:Blake & Zisserman functional

F (K0,K1, v) =

=

∫Ω\(K0∪K1)

(∣∣D2v(x)∣∣2

+ |v(x)− g(x)|2)

dx +

+ α Hn−1(K0) + β Hn−1(K1 \ K0)

(2)

to be minimized among admissible triplets ( K0, K1, v ) :

K0 , K1 closed subsets of Rn,

u ∈ C2(Ω \ (K0 ∪ K1)) and continuous on Ω \ K0.

with data :

Ω ⊂ Rn open set, n ≥ 1,

g ∈ L2(Ω) grey level intensity of the given image,

α, β positive parameters(chosen accordingly to scale and contrast threshold),

Hn−1 denotes the (n − 1) dimensional Hausdorff measure.

Franco Tomarelli


Existence of minimizers for (2) has been proven by

Coscia n = 1(strong and weak form. coincide iff n = 1 !), and by

[Carriero, Leaci & T.] n = 2,

via direct method in calculus of variations:solution of a weak formulation of minimum problem(performed for any dimension n ≥ 2)and subsequently proving additional regularityof weak minimizers under Neumann bdry condition (n = 2)

[C-L-T, Ann.S.N.S., Pisa (1997)]

Since we looked for a weak formulationof a free discontinuity problem,we wrote a suitable relaxed form relaxed version of BZ functional;this form depends only on u (not on triplets!):optimal segmentation (K0 ∪ K1) has to be recovered throughjumps (u discontinuity set) and creases (Du discontinuity set)

[C-L-T, in PNLDE, 25 (1996)]

Franco Tomarelli


We proved also several density estimatesfor minimizers energy and optimal segmentation:

[C-L-T, Nonconvex Optim. Appl.55 (2001)],

[C-L-T, C.R.Acad.Sci.(2002)],

[C-L-T J. Physiol.(2003)];

by exploiting this estimates, via Gamma-convergence techniques,

[Ambrosio, Faina & March, SIAM J.Math.An. (2002)]obtained an approximation of Blake & Zisserman functionalwith elliptic functionals,

and numerical implementation was performed by

[R.March ]

[M.Carriero, A.Farina, I.Sgura ].

Franco Tomarelli


No uniqueness due to nonconvexity,

nevertheless generic uniqueness olds true in 1-D.

About uniqueness and well-posedness:

[T.Boccellari, F.T., Ist.Lombardo Rend.Sci 2008, 142237-266] (n ≥ 1),

[T.Boccellari, F.T.] QDD Dip.Mat.Polit.MI 2010] (n = 1),

Franco Tomarelli


Stime a priori e continuità del valore di minimo

Theorem - Minimizing triplets (K0,K1,u) ofBlake & Zisserman F g

α,β functional fulfil (in any dimension n):

‖u‖L2 ≤ 2 ‖g‖L2 ,

0 ≤ mg(α, β) ≤ ‖g‖2L2 ,∣∣mg(α, β)−mh(a,b)∣∣ ≤ 5(‖g‖L2 + ‖h‖L2) ‖g − h‖L2 +

min‖g‖2

L2 , ‖h‖2L2

minα,a

|α− a|+min

‖g‖2

L2 , ‖h‖2L2

minβ,b

|β − b| ,

Hn−1(K0) ≤2α

(‖g‖2 + η2

), Hn−1(K1 \ K0) ≤

2β

(‖g‖2 + η2

)per ogni terna (u,K0,K1) minimizzante F h

α,β con ‖h − g‖L2 < η .

Franco Tomarelli


notice that 1-dimensional case fits very well to a short presentation,since (only in 1-d) strong and weak functional coincide.

1-d Blake & Zisserman 1-d functional

Given g ∈ L2(0,1), α, β ∈ R we set F gα,β :

F gα,β(u) =

∫ 1

0|u(x)|2 dx+

∫ 1

0|u(x)− g(x)|2 dx+α ] (Su)+β ] (Su\Su)

(3)to be minimized among u ∈ L2(0,1) t.c. ] (Su ∪ Su)<+∞ t.c.u′,u′′ ∈ L2(I) for every interval I ⊆ (0,1)\(Su ∪ Su)

Notation :u denotes the absolutely continuous part of u′,u the absolutely continuous part of (u)′ = u′′,Su ⊆ (0,1) the set of jump points of u,Su ⊆ (0,1) the set of jump points of u,] the counting measure.

Franco Tomarelli


n = 1

Summary of analytic results:

Euler equations for local minimizers,

compliance identity for local minimizers,

a priori estimates on minimum value and minimizers,

continuous dependence of minimum value mg(α, β) with respectto g, α, β.

Theorem

F gα,β achieves its minimum provided

the following conditions are fulfilled:

0 < β ≤ α ≤ 2β < +∞ (4)

g ∈ L2. (5)

Uniqueness fails

Franco Tomarelli


There are many kinds of uniqueness failure:

precisely, even considering the simple 1-d case:

if g has a jump,then there ∃ α > 0 s.t . F g

α,α has exactly two minimizers;

there are α > 0 and g ∈ L2(0,1) s.t. uniqueness fail for every β ina non empty interval (α− ε, α];

for every α and β fulfilling 0 < β ≤ α < 2β there is g ∈ L2(0,1)s.t.

] (argmin F gα,β) ≥ 2.

Eventually we can show an example of a setN ⊆ L2(0,1) with non empty interior part in L2(0,1)s.t. for every g ∈ N there are α and β satisfying (4) and

] (argmin F gα,β) ≥ 2.

Franco Tomarelli


Generic uniqueness

Euler eqs are an over-determined system(singular set is an unknown)

Nevertheless we can prove

Theorem ([T.Boccellari & F.T ])

For any α, β s.t.

0 < β ≤ α ≤ 2β , α/β 6∈ Q ,

there is a Gδ (countable intersection of dense open sets)set Eα,β ⊂ L2(0,1) such that

] (argminF gα,β) = 1 .

Franco Tomarelli


Idea of the proof:

we show analytic dependence of(absolutely continuous part of) energywith respect to variations of open cells ofCW-complex structure of partitions of (0,1)induced by singular set of piecewise affine data g

the set of all piece-wise affine data(related to suitably refined partitions of (0,1))and exhibiting non uniqueness of minimizer withdifferent quality” (ordering of jump and creases)and same prescribed cardinality of singular set has null mdimensional Lebesgue measure(here m is the dimension of the space of continuous piece-wisefunctions in (0,1) affine with at most m creases)

technical density argument

Franco Tomarelli


The whole picture is coherent with the presence of instable patterns,each of them corresponding to a bifurcation of optimal segmentationunder variation of parameters α e β , related to:

contrast threshold (√α ),

“luminance sensitivity”,

resistance to noise,

crease detection (√β ),

double edge detection.

Franco Tomarelli


Dirichlet problem for BZ functional, n = 2

Image InPainting refers to the filling in of missing or partially occludedregions of an image.

Minimizing Blake & Zisserman functional is usefulto achieve contour continuation in the whole image region Ω

when occlusion or local damage occur in Ω \ Ωe.g. blotches in a fresco or a movie film.

Dirichlet problem:minimize the energy F (K0,K1, v) in Ω ⊂ R2:

F (K0,K1, v) =

=

∫eΩ\(K0∪K1)

(∣∣D2v(x)∣∣2

+ µ |v(x)− g(x)|2)

dx +

+ α H1(K0) + β H1(K1 \ K0)

(6)

among triplets which assume prescribed data w on Ω \ Ω:say v = w a.e. Ω \ Ω

Franco Tomarelli


Weak formulation of Dirichlet pb for BZ functional

Minimize F : X → [0,+∞] defined by

F(v) =

∫eΩ

(|∇2v |2 + µ|v − g|2) dx + αH1(Sv ) + βH1(S∇v \ Sv ) (7)

where Ω ⊂⊂ Ω ⊂ R2 are open sets, x = (x , y) ∈Omegaand

X = GSBV 2(Ω) ∩ L2(Ω) ∩

v = w a.e.Ω \ Ω

.

Theorem

If g ∈ L2(Ω), w ∈ X and β ≤ α ≤ 2βthen F has at least one minimizer in X .

The main part F is denoted by E :

E(v) =

∫eΩ

|∇2v |2 dx + αH1(Sv ) + βH1(S∇v \ Sv ) (8)

Franco Tomarelli


the example with cut and tilted disks tells that

1 sublevels of functional E are not compact onv ∈ SBV (Ω) : ∇v ∈ SBV (Ω)2

2 by letting untilted some of the big disks

we find functions with unbounded gradientwith arbitrarily small energy E

Franco Tomarelli


We recall the definitions of some function spaces related to firstderivatives which are special measures in the sense of De Giorgi

SBV (Ω) denotes the class of functions v ∈ BV (Ω) s.t.∫Ω

|Dv | =∫

Ω

|∇v |dy +

∫Sv

|v+ − v−|dH1.

SBVloc(Ω) = v ∈ SBV (Ω′) : ∀Ω′ ⊂⊂ Ω ,

GSBV (Ω) =

v : Ω → R Borel ; −k ∨ v ∧ k ∈ SBVloc(Ω) ∀k

GSBV 2(Ω) =

v ∈ GSBV (Ω), ∇v ∈(GSBV (Ω)

)2

Franco Tomarelli


We emphasize that

GSBV (Ω), GSBV 2(Ω) are neither vector spacesnor subsets of distributions in Ω.

Neverthelesssmooth variations of a function in GSBV 2(Ω)still belong to the same class.

Notice that,

if v ∈ GSBV (Ω), then Sv is countably (H1,1) rectifiable and ∇vexists a.e. in Ω.

Dv 6= ∇v in GSBV 2(Ω)

S∇v =⋃2

i=1 S∇i v

Franco Tomarelli


Remark

1 v ∈ BV ∩ L∞ , P(E) < +∞ ⇒ v χE ∈ BV

2 v ∈ BV , P(E) < +∞ 6⇒ ∗ v χE ∈ BV

3 v ∈ BV , P(E) < +∞ ⇒ v χE ∈ GBV

∗ the trace of v could be not integrable, e.g.:

n = 2 Ω = B1 v = %−1/2 ∈ W 1,1(B1)

E =

x = x , y 1

k2 + 1< % <

1k2 , k ∈ N

Franco Tomarelli


Theorem [CLT, Adv.Math.Sci.Appl., 2010]existence of strong minimizer

Assume

0 < β ≤ α ≤ 2β, µ > 0, g ∈ L2(Ω) ∩ L4loc(Ω) , w ∈ L2(Ω) ,

Ω is a bounded open set with C2 boundary ∂Ω ,

w ∈ C2(Ω

), D2w ∈ L∞(Ω ) .

Then there is at least one triplet (C0,C1,u) minimizing functional

F (K0,K1, v) =

∫Ω\(K0∪K1)

(∣∣D2v(x)∣∣2

+ |v(x)− g(x)|2)

dx +

+ α H1(K0) + β H1(K1 \ K0)with finite energy, among admissible triplets (K0,K1, v): K0 , K1 Borel subsets of R2, K0 ∪ K1 closed,

v ∈ C2 (Ωε \ (K0 ∪ K1)) , v approximately continuous in (Ωε \ K0) ,v = w a.e. in Ωε \ Ω .

Franco Tomarelli


Moreover, any minimizing triplet (K0,K1, v) fulfils:

K0 ∩ Ω and K1 ∩ Ω are (H1,1) rectifiable sets,

H1(K0 ∩ Ω) = H1(Sv ) , H1(K1 ∩ Ω) = H1(S∇v \ Sv ) ,

v ∈ GSBV 2(Ω), hence

v ∇v have well defined two-sided traces, H1 a.e. finite on K0 ∪ K1,

the function v is also a minimizer of the weak functional F

F(z) =

∫Ω

(|∇2z|2 + µ|z − g|2) dx + αH1(Sz) + βH1(S∇z \ Sz)

over z ∈ L2(Ω) ∩GSBV (Ω) : ∇z∈(GSBV (Ω))2, z = w a.e. Ω \ Ω.

Eventually, the third element v of any minimizing triplet (K0,K1, v)fulfils

F(v) = F (K0,K1, v) .

Franco Tomarelli


Steps of the proof

Existence of minimizing triplets is achieved by showing partialregularity of the weak solution with penalized Dirichlet datum. Thenovelty consists in the regularization at the boundary for a freegradient discontinuity problem;

regularity is proven at points with 2-dimensional energy density by:

1 blow-up technique2 suitable joining along lunulae filling half-disk3 a decay estimate for weak minimizers

In the blow-up procedure, two refinements of relevant tools are

hessian decay of a function which is bi-harmonic inhalf-disk and vanishes together with normal derivativeon the diameter

a Poincaré-Wirtinger inequality forGSBV functions vanishing in a sector

Franco Tomarelli


Theorem (Biharmonic extension and L2 decay of Hessian)

Set B+R = BR(0) ∩ y > 0 ⊂ R2 , R > 0 .

Assume z ∈ H2(B+R ) , ∆2z ≡ 0 B+

R , z = zy ≡ 0 on y = 0.)

Then there exists an (obviously unique) extension Z of z in whole BR

such that ∆2Z ≡ 0 BR .

This extension may increase a lot L2 hessian norm of D2Znevertheless it implies nice decay on half-ball:

‖D2Z‖2L2(B+

ηR)≤ η2‖D2z‖2

L2(B+R ).

Such extimate is not a straightforward consequence of classicalSchwarz reflection principle for harmonic functions vanishing on thediameter, since the Almansi decomposition on the half-disk B+

R mayneither respect the vanishing value on the diameter:e.g. %3

(cosϑ− cos(3ϑ)

)= %2ϕ+ ψ where ϕ = x , ψ = 3x2y − x3

are both harmonic but do not vanish on the diameter y = 0,nor preserves orthogonality in L2 or H2:cancelation of big normsmay take place in one half-disk and not in the other: see Fig. (39)

Franco Tomarelli


Duffin extension formula

Assumez ∈ H2(B+

1 ),z is bi-harmonic in B+

1z = ∂z/∂y = 0 on B1(0) ∩ y = 0.

Thenz has a bi-harmonic extension Z in B1 defined by

Z (x , y) = z(x , y) ∀ (x , y) ∈ B+

1 ,Z (x ,−y) = −z(x , y) + 2yzy (x , y)− y2∆z(x , y) ∀ (x ,−y) ∈ B−1 .

Franco Tomarelli


Almansi-type decomposition (revisited)

Let u ∈ H2(BR \ Γ).

Then∆x

2u = 0 BR \ Γ (9)

iff

∃ ϕ,ψ : u(x) = ψ(x) + ‖x‖2 ϕ(x), ∆x ϕ(x) = ∆x ψ(x) ≡ 0, BR \ Γ.(10)

Moreover decomposition (10) is uniqueup to possible linear terms in ψ:

say A% cosϑ = Ax and B% sinϑ = Bythat can switch independently torespectively A%−1 cosϑ and B%−1 sinϑ in ϕ.

Back to hessian decay estimate (36)Franco Tomarelli


-1.0-0.5

0.00.5

1.0

-1.0

-0.5

0.0

0.5

1.0

-2

0

2

Franco Tomarelli


We use a new Poincaré-Wirtinger type inequality in the class GSBVwhich allows surgical truncations of non integrable functions ofseveral variables with the aim of taming blow-up at boundary points incase of functions vanishing in a full sector.

Notice that v ∈ GSBV 2(Ω) does not even entail that either v or ∇vbelongs to L1

loc(Ω).

Franco Tomarelli


DECAY

Theorem - Decay of functional F at boundary pointsThere are constants C1, C2 (dep. on α, g, w ) s.t.,

∀k > 2, ∀η, σ ∈ (0,1) with η < C2

∃ε0 > 0, ϑ0 > 0 : ∀ε ∈ (0, ε0],

∀ x ∈ ∂Ω and any local minimizer u of F in Ω ∩ B%(x) , s.t.

0 < % ≤(εk ∧ C1

),

∫B%(x)

|g|4 ≤ εk and

αH1(

Su ∩ Ω ∩ B%(x))

+ βH1((S∇u \ Su) ∩ Ω ∩ B%(x)

)< ε% ,

we have

FBη%(x)(u) ≤

≤ η2−σ maxFB%(x)(u) , %2 ϑ0

((Lip(γ∂Ω

′))2

+(Lip(Dw)

)2)

.

Franco Tomarelli


Admissible triplets and localizationAdmissible triplets: (K0,K1, v) is an admissible triplet if

K0 , K1 Borel subsets of R2, K0 ∪ K1 closed,

v ∈ C2(Ω \ (K0 ∪ K1)

), v approximately continuous in Ω \K0,

v = w a.e. in Ω \ Ω .

(Localization)

We will use the symbols FA, FA to denote respectively functionals F ,F when Ω is substituted by a Borel set A ⊂ Ω, (resp. EA,FA for E ,F)

(Locally minimizing triplet of F (6))

Admissible triplet (K0,K1,u), is a locally minimizing triplet of F if

FA(K0,K1,u) < +∞

FA(K0,K1,u) ≤ FA(T0,T1, v)

∀ smooth open A ⊂⊂ Ω and any admissible triplet (T0,T1, v) s.t.

spt(v − u) and (T0 ∪ T1)4(K0 ∪ K1) are subsets of A.Franco Tomarelli


(Essential locally minimizing triplet of F , resp. E)

Given a locally minimizing triplet (T0,T1, v) of the functional F(resp. E),there is another triplet (K0,K1,u) , called

essential locally minimizing triplet , which is uniquely defined by

u = v

K0 = T0 ∩ K \ (T1 \ T0)

K1 = T1 ∩ K \ T0

where v is the approximate limit of v , a.e. defined by

g(v(x)) = lim%↓0

∫Bρ(0)

g(v(x + y))dy ∀g ∈ C0(R)

and K isthe smallest closed subset of T0 ∪ T1 such that v ∈ C2(Ω \ K ).

Franco Tomarelli


Some Euler equations in 2 dimensional case[C.L.T] Calc.Var.Part.Diff.Eq, 2008[C.L.T] Prepr.24 Dip.Mat.Univ.Salento, 2008

∆2u + µu = µg Ω \ (K0 ∪ K1)

Neumann boundary operators (plate-type bending moments)vanishing in K0 ∪ K1[[|D2u|2 + µ|u − g|2

]]= α K(K0)

[[|D2u|2

]]= β K(K1 \ K0)

Integral and geometric conditionsat the “boundary” of singular set: crack-tip and crease-tip.

Franco Tomarelli


Euler equations

From now on, for sake of simplicity,we examine only the main part E of functional F :

E(K0,K1, v) =

=

∫Ω\(K0∪K1)

∣∣D2v(x)∣∣2

dx + α H1(K0) + β H1(K1 \ K0)(11)

and the structural assumption β ≤ α ≤ 2β will be always understood.

Franco Tomarelli


Euler equations I : smooth variations

Theorem

Any essential locally minimizing triplet (K0,K1,u) for functional Ffulfils

∆2u + µ (u − g) = 0 in Ω \ (K0 ∪ K1) .

Any essential locally minimizing triplet (K0,K1,u) for the functional Efulfils

∆2u = 0 in Ω \ (K0 ∪ K1) .

Franco Tomarelli


Euler equations II :boundary-type conditions on singular set

Necessary conditions on jump discontinuity set K0

for natural boundary operators

Assume (K0,K1,u) is an essential locally minimizing triplet for thefunctional E , B ⊂⊂ Ω is an open disk such that K0 ∩B is a diameter ofthe disk and (K1 \ K0) ∩ B = ∅. Then(

∂2u∂N2

)±= 0 on K0 ∩ B ,

(∂3u∂N3 + 2

∂

∂N

(∂2u∂τ2

))±= 0 on K0 ∩ B

where B+,B− are the connected components of B \ K0, N is the unitnormal to K0 pointing toward B+, v+, v− the traces of any v on K0

respectively from B+ and B−,τ = (τ1, τ2) = (−N2,N1) the choice ofthe unit tangent vector to K0 .

Franco Tomarelli


Euler equations III : singular set variations

Next we evaluate the first variation of the energyaround a local minimizer u,under compactly supported smooth deformation of K0 and K1

Integral Euler equation

If (K0,K1,u) is a locally minimizing triplet of E . Then ∀η ∈ C20(Ω,R2)∫

Ω\(K0∪K1)

(|D2u|2 div η − 2

(DηD2u + (Dη)t D2u + Du D2η

): D2u

)dx

+ α

∫K0

divτK0η dH1 + β

∫K1\K0

divτK1\K0

η dH1 = 0 ,

where divτS denotes the tangential (to set S) divergence and(

DηD2u + (Dη)tD2u + DuD2η)

ij =

=∑

k

(DkηiD2

kju + Diηk D2kju + Dk uD2

ij ηk

)Franco Tomarelli


Curvature of jump set K0 and squared hessian jump

If (K0,K1,u) is an essential locally minimizing triplet for functional E ,B ⊂⊂ U ⊂ Ω two open disks, s.t. K0 ∩ U is the graph of a C4 function,B+ (resp. B−) the open connected epigraph (resp. subgraph) of suchfunction in B,.K1 ∩ U = ∅, and u in W 4,r (B+) ∩W 4,r (B−), r > 1 .

Then [[|D2u|2

]]= α K(K0) on K0 ∩ B .

where we denoteby K the curvature and by

[[w

]]the jump of a function w on K0

Analogous results holds true for crease set K1 \ K0

Both results follows by plugging(normal to singular set) vector fields in Integral Euler equation

Franco Tomarelli


Crack-tipNow we perform a qualitative analysis of the “boundary” of thesingular set, by assuming it is manifold as smooth as required by thecomputation of boundary operators.The strategy is a new choice of the test functions in Euler equation: avector field η tangential to K0 (or K1).

Crack-tip Theorem

Assume (K0,K1,u) is an essential locally minimizing triplet of E ,B = B(x0) ⊂ Ω an open disk with center at x0 s.t. (K1 \ K0) ∩ B = ∅ ,K0 ∩ B = Su ∩ B is a is a smooth curve from center to bdry of B and

∃ r > 1 : u ∈ W 4,r(U \ (K0 ∪ Bε(x0))

∀ ε > 0

Then u fulfils, for every η ∈ C30(B,R2) s.t. η = ζτ

( ζ ∈ C∞0 (B), τ ∈ C3(B,S1) and

η vector field tangent to K0 pointing toward K0 at x0)

limε→0+

∫∂Bε(x0)\K0

Lη(u) dH1 = α ζ(x0)

Franco Tomarelli


Q

δε ,Q

εΓ

Γη

εn

ε

δ

ν

εΞ

Ξ

uS

Franco Tomarelli


Summarizing:

By performing suitable smooth variationswe found Euler equation in Ω \ K0 ∪ K1) and jump conditions foru and for Du in K0 ∪ K1;

by performing smooth variations of jump and crease setsK0, K1 \ K0 around a minimizer we found integral and geometricconditions on optimal segmentation sets.

In addition we proved:

Franco Tomarelli


1 Caccioppoli inequality: as a consequenceany locally minimizing triplet of E in R2 with finite energy andcompact segmentation set K0 ∪ K1

actually must have empty segmentation;2 Liouville property:

(∅, ∅,u) with bi-harmonic u is locally minimizing triplet of E in Rn

iff u is affine;3 neither a straight infinite wedge

nor a straight 1–dimensional uniform jumpare locally minimizing triplets of E in R2;

4 3/2 homogeneity: .any locally minimizing triplet (K0,K1,u) is transformed in anotherlocally minimizing triplet by allnatural re-scaling centered at x0 ∈ Ω, which maps

u(x) to %−3/2u(x0 + %x) ,

Kj to %−1(Kj − x0), % > 0, j = 0,1 .

Franco Tomarelli


Mode 1 (JUMP) :

%3/2 ω(θ) = %3/2(

sinθ

2− 5

3sin

(32θ

))− π < θ < π

Mode 2 (CREASE) :

%3/2 w(θ) = %3/2(

cosθ

2− 7

3cos

(32θ

))− π < θ < π

CANDIDATE:

W = ±√

α

193π%3/2

(√21ω(θ) ± w(θ)

)− π < θ < π

W fulfils all Euler equations,all constraints on jump and curvature of singular set and

Energy equipartition:∫

B%(0)

|∇2u|2 dx = α%

Franco Tomarelli


Candidate conjecture

Assume 0 < β ≤ α ≤ 2β < +∞.

Then triplet

( K0 = negative real axis , K1 = ∅ , function W ) )

is a locally minimizing triplet for E in R2 .

Moreover we conjecture thatthere are no other nontrivial locally minimizing tripletswith non empty jump set and different from triplets

(K0 = closed negative real axis, K1 = ∅, Φ)

Φ = (Aω(ϑ) + B w(ϑ)) r3/2, 35 A2 + 37 B2 =4απ, A 6= 0

possibly swayed by rigid motions of R2 co-ordinatesand/or addition of affine functions.

(69)Franco Tomarelli


Proving the minimality of a given candidate for a free discontinuityproblem is a difficult task in general.

As far as we know,neither the calibration techniques [Alberti, Bouchitte,DalMaso], nor the method used by [Bonnet, David] (bothsuccessfully applied to Mumford & Shah functional to test non trivialminimizers) seem to apply to the present context of second orderfunctionals.

Even the excess identity approach of [Percivale & T.],which succeeds with second order functionals related toelasto-plastic plates,does not apply to the present context since Blake & Zissermanfunctional do not control

∫SDv|[Dv ]|dH1.

Franco Tomarelli


Mumford-Shah functional

Theorem [M.Carriero, A.Leaci, D.Pallara, E.Pascali]

If (R−,u) is a local minimizer of∫B1

|∇v |2 + αH1(Sv )

thenu(ρ, θ) = a0 ± uS(ρ, θ) + uR(ρ, θ))

where

uS(ρ, θ) =

√2απρ1/2 sin

θ

2, uR(ρ, θ) = o(ρ1−ε)

Franco Tomarelli


CONJECTURE ( E.De Giorgi )

ψ(ρ, θ) =

√2απρ1/2 sin

θ

2

is a local minimizer of Mumford-Shah functional in R2.ψ is the only non trivial local minimizer in R2

(up to the sign and/or a rigid motion and constant addition)

where local minimizer of M–S functional refers tocompactly-supported variation(without topological restrictions)

With a slightly different definitioncompetitor for (u,K ): any pair (w ,H) s.t. ....... ...... andif x , y ∈ R2 \ (K ∪ BR) are separated by K ,then also H separates them,

A.Bonnet & G.David proved the conjecture in a weak form.(the difference does not play any role for candidate ψ.)

Franco Tomarelli


Theorem - Uniform density estimates up to the bdry[C-L-T, Pure Math.Appl., 2009]

(Density upper bound for the functional F )Let (K0,K1,u) be an essential locally minimizing triplet for thefunctional F under structural assumptions, g ∈ L4

loc(Ω), and

∃% > 0 : H1(∂Ω ∩ B%(x))< C% ∀x ∈ ∂Ω , ∀% ≤ % .

Then for every 0 < % ≤ (% ∧ 1) and for every x ∈ Ω such thatB%(x) ⊂ Ω we have

F B%(x)∩Ω (K0,K1,u) ≤ c0%

where c0 = C2π + 2π12µ

(‖w‖2

L4(B%(x)) + ‖g‖2L4(B%(x))

)+ (2π + C)α.

Franco Tomarelli


Theorem - Uniform density estimates up to the bdry[C-L-T, Comm.Pure Appl.Anal. 2010]

Let (K0,K1,u) be an essential locally minimizing triplet for thefunctional F under structural assumptions, g ∈ L4

loc(Ω), and

∃% > 0 : H1(∂Ω ∩ B%(x))< C% ∀x ∈ ∂Ω , ∀% ≤ % .

(Density lower bound for the functional F )Then there exist ε0 > 0, %0 > 0 such that, for every 0 < % ≤ (% ∧ 1)

and for every x ∈ Ω such that B%(x) ⊂ Ω we have

FB%(x)(K0,K1,u) ≥ ε0% ∀x ∈ (K0 ∪ K1) ∩ Ω, ∀% ≤ %0

(Density lower bound for the segmentation length )and there exist ε1 > 0, %1 > 0 such that

H1 ((K0 ∪ K1) ∩ B%(x)) ≥ ε1% ∀x ∈ (K0 ∪ K1) ∩ Ω, ∀% ≤ %1.

Franco Tomarelli



(Elimination property ) Let (K0,K1,u) be an essential locallyminimizing triplet for the functional F under structural assumptions,g ∈ L4

loc(Ω), and

∃% > 0 : H1(∂Ω ∩ B%(x))< C% ∀x ∈ ∂Ω , ∀% ≤ % .

Then and let ε1 > 0, %1 > 0 as above. If x ∈ Ω and

H1 ((K0 ∪ K1) ∩ B%(x)) <ε1

2%

then(K0 ∪ K1) ∩ B%/2(x) = ∅ .

Franco Tomarelli



(Minkowski content of the segmentation )Let (K0,K1,u) be an essential locally minimizing triplet for thefunctional F under structural assumptions and g ∈ L4(Ω).

Then K0 ∪ K1 is (H1,1) rectifiable and

lim%↓0

|x ∈ Ωε ; dist(x, (K0 ∪ K1) ∩ Ω) < % |2%

= H1 (K0 ∪ K1) .

Franco Tomarelli


Numerical experiments

[F.Doveri] proved the Γ convergence and implemented the GNCalgorithm proposed by Blake & Zisserman.[G.Bellettini, A.Coscia] (n=1) approximation by elliptic functionals.[L.Ambrosio, L.Faina, R.March] (n=2) variational approximationof B& Z fctl:

Fε(u, s, σ) =

∫Ω

((σ2 + κε)|∇2u|2) + µ|u − g|2d x +

+(α− β)Gε(s) + β Gε(σ) + ξε

∫Ω

(s2 + ξε)|∇u|γdx

with κε, ξε, ζε suitable infinitesimal weights and

Gε(s) =

∫Ω

(ε |∇s|2 +

(s − 1)2

4ε

)dx

[M.Carriero, I.Farina, A.Sgura] implemented a finite differenceapproach via Euler-Lagrange equations.

Franco Tomarelli


Franco Tomarelli


Franco Tomarelli


Franco Tomarelli


Franco Tomarelli


Franco Tomarelli


Theorem - Asymptotic expansion of loc.min. triplets with crack-tip

Assume (Γ, ∅,u) is a locally minimizing triplet of E in R2 ,where Γ = denotes the closed negative real axis.Then there are constants A, B with (A,B) 6= (0,0) and Ah,Bh s.t.

u(r , θ) =

= r3/2(

A(

sin(

θ2

)− 5

3 sin(

32θ

))+ B

(cos

(θ2

)− 7

3 cos(

32θ

)))+

++∞∑h=1

rh+ 32

(Ah cos

((h + 3

2

)θ)

+ Bh sin((

h + 32

)θ)

+

− 2h+32h+7 Ah cos

((h − 1

2

)θ)− 2h+3

2h−5 Bh sin((

h − 12

)θ))

where u is expressed by polar coordinates in R2

with θ ∈ (−π, π) and r ∈ (0,+∞) .

This expansion is strongly convergent in H2(B% \ Γ), moreover ...

Franco Tomarelli


... the lower order term (h = 0) in the expansionmust have the following form

W0 = (Aω(ϑ) + B w(ϑ)) r3/2 in B% \ Γ , referring to modes :

Mode 1 (Jump) ω(ϑ) =

(sin

(ϑ

2

)− 5

3sin

(32ϑ

))

Mode 2 (Crease) w(ϑ) =

(cos

(ϑ

2

)− 7

3cos

(32ϑ

))where ϑ ∈ (−π, π) and constants A, B verify

35 A2 + 37 B2 =4απ, A 6= 0 .

Franco Tomarelli

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FREE GRADIENT DISCONTINUITY AND IMAGE SEGMENTATIONansobol/otarie/slides/tomarelli.pdf · This talk deals with free discontinuity problems related to contour enhancement in image segmentation,